Generalized Geometric Programming Research Articles

Global Optimization in Robust Fractional Control of Uncertain Fractional Order Systems: A Thermal Application Using the STM32 Microcontroller

In this paper, we suggest an improvement to our previously undertaken approach. Briefly, this approach consisted of applying the robust fractional predictive control (RFPC) for a class of constrained fractional systems implementing the min–max optimization technique. The RFPC controller requires resolution of a non-convex min–max optimization problem. The resolution of this problem, however, can only conduce to local solutions. The reason is simple: the objective function to be optimized is non-convex due to the presence of uncertainties. In the present work, we propose a global optimization-based RFPC controller for an uncertain fractional order system. A determinist global optimization method, namely, generalized geometric programming (GGP), is proposed to solve this problem for the uncertain fractional order system. The GGP method consists of converting a non-convex problem into a convex one via the application of variable changes. The technique of the convexification of this method is applied in line with the objective function to be optimized. Consequently, we obtained a new convex criterion and a convex problem. From an experimental point of view, we applied the proposed RFPC to a real thermal system using an STM32 microcontroller in order to control our thermal system.

Efficient Convexification Strategy for Generalized Geometric Programming Problems

Generalized geometric programming (GGP) problems consist of a signomial being minimized in the objective function subject to signomial constraints, and such problems have been utilized in various fields. After modeling numerous applications as GGP problems, solving them has become a significant requirement. A convex underestimator is considered an important concept to solve GGP problems for obtaining the global minimum. Among convex underestimators, variable transformation is one of the most popular techniques. This study utilizes an estimator to solve the difficulty of selecting an appropriate transformation between the exponential transformation and power convex transformation techniques and considers all popular types of transformation techniques to develop a novel and efficient convexification strategy for solving GGP problems. This proposed convexification strategy offers a guide for selecting the most appropriate transformation techniques on any condition of a signomial term to obtain the tightest convex underestimator. Several numerical examples in the online supplement are presented to investigate the effects of different convexification strategies on GGP problems and demonstrate the effectiveness of the proposed convexification strategy with regard to both solution quality and computation efficiency. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0850 .

Robust Fixed Low-Order Controller for Uncertain Decoupled MIMO Systems

The design of a robust fixed low-order controller for uncertain decoupled multi-input multi-output (MIMO) systems is proposed in this paper. The simplified decoupling is used as a decoupling system technique. In this work, the real system behavior is described by a linear model with parametric uncertainties. The main objective of the control law is to satisfy, in presence of model uncertainties, some step response performances such as the settling time and the overshoot. The controller parameters are obtained by resolving a min–max nonconvex optimization problem. The resolution of this kind of problems using standard methods can generate a local solution. Thus, we propose, in this paper, the use of the generalized geometric programming (GGP) which is a global optimization method. Simulation results and a comparison study between the presented approach, a proportional integral (PI) controller, and a local optimization method are given in order to shed light the efficiency of the proposed controller.

Synthesis and Implementation of a Fixed Low Order Controller on an Electronic System

The development of microelectronic field and software allows researchers to implement some control law in miniaturized devices such as Field Programmable Gate Arrays (FPGA) and microcontroller. These control laws may be used in industrial applications. The key of this work is the design and the implementation of a fixed low order controller on a STM32 microcontroller in order to control an electronic system. The main objective of this controller is to ensure some time response performances as the settling time and the overshoot. The controller parameters are obtained by resolving a non convex optimization problem while considering the desired closed loop specifications. So, the use of a classical optimization method to resolve such kind of problems may lead to a local solution and then the obtained solution is not optimal. Therefore, it is suggested to apply a global optimization method in order to get an optimal control law that can ensure the specified time response performances. The proposed method in this work is the Generalized Geometric Programming (GGP) method. This method consists on transforming, by some mathematical transformations, a non convex optimization problem to a convex one. The implementation of a Proportional Integral (PI) controller, a Proportional Integral Derivate (PID) and a fixed low order controller, on a real electronic system, shows the efficiency of the latter one.

Solving Continuous Network Design Problem with Generalized Geometric Programming Approach

To satisfy growing travel demand and reduce traffic congestion, the continuous network design problem (CNDP) is often proposed to optimize road network performance by the expansion of road capacity. In the determination of the equilibrium travel flow pattern, equilibrium principles such as deterministic user equilibrium (DUE) and stochastic user equilibrium (SUE) may be applied to describe travelers’ route choice behavior. Because of the different mathematical formulation structures for the CNDP with DUE and SUE principles, most of the existing solution algorithms have been developed to solve the CNDP for either DUE or SUE. In this study, a more general solution method is proposed by applying the generalized geometric programming (GGP) approach to obtain the global optimal solution of the CNDP with both DUE and SUE principles. Specifically, the original CNDP problem is reformulated into a GGP form, and then a successive monomial approximation method is employed to transform the GGP formulation into a standard geometric programming form, which can be cast into an equivalent nonlinear but convex optimization problem whose global optimal solution can be guaranteed and solved by many existing solution algorithms. Numerical experiments are presented to demonstrate the validity and efficiency of the solution method.

A MILP formulation for generalized geometric programming using piecewise-linear approximations

Generalized geometric programming (GGP) problems are converted to mixed-integer linear programming (MILP) problems using piecewise-linear approximations. Our approach is to approximate a multiple-term log-sum function of the form log (x1 + x2 + ⋅⋅⋅ + xn) in terms of a set of linear equalities or inequalities of log x1, log x2, …, and log xn, where x1, …, xn are strictly positive. The advantage of this approach is its simplicity and readiness to implement and solve using commercial MILP solvers. While MILP problems in general are no easier than GGP problems, this approach is justified by the phenomenal progress of computing power of both personal computers and commercial MILP solvers. The limitation of this approach is discussed along with numerical tests.

Synthesis of Controllers for MIMO Systems with Time Response Specifications

This paper proposes the design of fixed low order controllers for Multi Input Multi Output (MIMO) decoupled systems. The simplified decoupling is used as a decoupling system technique due to its advantages compared to other decoupling methods. The main objective of the proposed controllers is to satisfy some desired closed loop step response performances such as the settling time and the overshoot. The controller design is formulated as an optimization problem which is non convex and it takes in account the desired closed loop performances. Therefore, classical methods used to solve the non convex optimization problem can generate a local solution and the resulting control law is not optimal. Thus, the thought is to use a global optimization method in order to obtain an optimal solution which will guarantee the desired time response specifications. In this work the Generalized Geometric Programming (GGP) is exploited as a global optimization method. The key idea of this method consists in transforming an optimization problem, initially, non convex to a convex one by some mathematical transformations. Simulation results and a comparison study between the presented approach and a Proportional Integral (PI) controller are given in order to shed light the efficiency of the proposed controllers.

Cooperative Secure Resource Allocation in Cognitive Radio Networks with Guaranteed Secrecy Rate for Primary Users

In this paper, we introduce a new cooperative paradigm for secure communication in cognitive radio networks (CRNs) where secondary users (SUs) are allowed to access the spectrum of primary users (PUs) as long as they preserve the secure communication of PUs in the presence of malicious eavesdroppers. To do so, the SU transmission is divided into two hops: at first hop, the SU transmitter sends the information to a relay set and the SU receiver acts as a friendly jammer to disturb the overhearing of eavesdroppers and at the second hop, one of the relays is selected to pass the information to the SU receiver and the SU transmitter acts as a friendly jammer for the PU. In this new setup, the time duration for each hop, the power transmissions of all nodes in CRN, and relay selection at the second hop are allocated in such a way that the secrecy rate of the SU is maximized subject to the minimum required PU's secrecy rate. From primary service perspective, this transforms the possibly disturbing secondary service activities into a beneficial network element. We investigate instantaneous and ergodic resource allocation problems for perfect and imperfect channel state information (CSI). Since these problems are non-convex, we propose a solution based on decomposition of main optimization problem into three subproblems related to the power allocation, time allocation, and relay selection. We show that the power allocation problem can be transformed into a generalized geometric programming (GGP) model via the so-called scaled algorithm and it can be solved very efficiently. Simulation results indicate that in terms of the secondary secrecy rate, the proposed setup outperforms the conventional setup in which the secrecy rate of the PU is not guaranteed.

Global optimization for generalized geometric programming problems with discrete variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.

A GGP approach to solve non convex min-max predictive controller for a class of constrained MIMO systems described by state-space models

This paper proposes a new method to solve non convex min-max predictive controller for a class of constrained linear Multi Input Multi Output (MIMO) systems. A parametric uncertainty state space model is adopted to describe the dynamic behavior of the real process. Moreover, the output deviation method is used to design the j-step ahead output predictor. The control law is obtained by the resolution of a non convex min-max optimization problem under input constraints. The key idea is to transform the initial non convex optimization problem to a convex one by means of variable transformations. To this end, the Generalized Geometric Programming (GGP) which is a global deterministic optimization method is used. An efficient implementation of this approach will lead to an algorithm with a low computational burden. Simulation results performed on Multi Input Multi Output (MIMO) system show successful set point tracking, constraints satisfaction and good non-zero disturbance rejection.

A New Global Optimization Algorithm for Solving Generalized Geometric Programming

A global optimization algorithm for solving generalized geometric programming (GGP) problem is developed based on a new linearization technique. Furthermore, in order to improve the convergence speed of this algorithm, a new pruning technique is proposed, which can be used to cut away a large part of the current investigated region in which the global optimal solution does not exist. Convergence of this algorithm is proved, and some experiments are reported to show the feasibility of the proposed algorithm.

Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables

Many optimization problems are formulated as generalized geometric programming (GGP) containing signomial terms f(X)·g(Y), where X and Y are continuous and discrete free-sign vectors, respectively. By effectively convexifying f(X) and linearizing g(Y), this study globally solves a GGP with a lower number of binary variables than are used in current GGP methods. Numerical experiments demonstrate the computational efficiency of the proposed method.

A global optimization using linear relaxation for generalized geometric programming

Many local optimal solution methods have been developed for solving generalized geometric programming (GGP). But up to now, less work has been devoted to solving global optimization of (GGP) problem due to the inherent difficulty. This paper considers the global minimum of (GGP) problems. By utilizing an exponential variable transformation and the inherent property of the exponential function and some other techniques the initial nonlinear and nonconvex (GGP) problem is reduced to a sequence of linear programming problems. The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of (GGP) on a microcomputer.

Treating free variables in generalized geometric programming problems

Generalized geometric programming (GGP) problems occur frequently in engineering design, chemical process industry and management. Although some exponential-based decomposition methods have been developed for GGP problems, these methods can only handle problems with positive variables. Applying GGP in real applications, free variables are usually introduced to model the system behavior or the management problems. Therefore, this study proposes a technique for treating free variables in GGP problems. Numerical examples are also presented to demonstrate the usefulness of the proposed method.

A robust algorithm for generalized geometric programming

Most existing methods of global optimization for generalized geometric programming (GGP) actually compute an approximate optimal solution of a linear or convex relaxation of the original problem. However, these approaches may sometimes provide an infeasible solution, or far from the true optimum. To overcome these limitations, a robust solution algorithm is proposed for global optimization of (GGP) problem. This algorithm guarantees adequately to obtain a robust optimal solution, which is feasible and close to the actual optimal solution, and is also stable under small perturbations of the constraints.

On generalized geometric programming problems with non-positive variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management. Some exponential-based decomposition methods have been developed for solving global optimization of GGP problems. However, the use of logarithmic/exponential transformations restricts these methods to handle the problems with strictly positive variables. This paper proposes a technique for treating non-positive variables with integer powers in GGP problems. By means of variable transformation, the GGP problem with non-positive variables can be equivalently solved with another one having positive variables. In addition, we present some computationally efficient convexification rules for signomial terms to enhance the efficiency of the optimization approach. Numerical examples are presented to demonstrate the usefulness of the proposed method in GGP problems with non-positive variables.

A new rectangle branch-and-pruning approach for generalized geometric programming

Generalized geometric programming (GGP) problem occurs frequently in engineering design and management. In this paper, a branch-and-pruning global optimization algorithm is proposed for GGP. By utilizing some transformations, a linear relaxation of the problem (GGP) is obtained based on the linear lower bound functions of objective and constraint functions inside some hyperrectangle region. Then a new pruning technique is given to accelerate the convergence of the given algorithm, and this pruning technique offers the possibility to cut away a large part of the current investigated region in which there no exist global optimum solution. The proposed algorithm which connects branch-and-bound method with the pruning technique successfully is convergent to the global minimum, according to the successive refinement of the linear relaxation of feasible region of the objective function and the solutions of a series of linear relaxation problems. And finally numerical experiment is given to illustrate the feasibility and efficiency of the proposed algorithm.

Treating Free Variables in Generalized Geometric Global Optimization Programs

Generalized geometric programming (GGP) problems occur frequently in engineering design and management. Recently, some exponential-based decomposition methods [Maranas and Floudas, 1997,Computers and Chemical Engineering 21(4), 351---370; Floudas et al., 1999 , Handbook of Test Problems in Local and Global Optimization, Kluwer Academic Publishers, Boston, pp. 5---105; Floudas, 2000 Deterministic Global Optimizaion: Theory, Methods and Application, Kluwer Academic Publishers, Boston, pp. 257---306] have been developed for GGP problems. These methods can only handle problems with positive variables, and are incapable of solving more general GGP problems. This study proposes a technique for treating free (i.e., positive, zero or negative) variables in GGP problems. Computationally effective convexification rules are also provided for signomial terms with three variables.

A deterministic global optimization algorithm for generalized geometric programming

A deterministic global optimization algorithm is proposed for generalized geometric programming (GGP). By utilizing some transformations, the initial non-convex problem is reduced to a reverse convex programming (RCP), where the objective function and constraint functions are convex. Then a linear relaxation of the problem (RCP) is obtained based on the linear lower bounding functions of the convex constraint functions and the linear upper bounding functions of the reverse convex constraint functions inside some hyperrectangle region. A cutting-plane method is proposed to add some effective linear constraints to the linear relaxation programming based on the famous arithmetic-geometric mean inequality, then derive a tighter linear relaxation programming. The proposed global optimization algorithm which connects the branch and bound method with the cutting-plane method successfully is convergent to the global minimum through the successive refinement of the linear relaxation of the feasible region of the objective function and the solutions of a series of linear relaxation problems. And finally the numerical experiment is given to illustrate the feasibility and the robust stability of the present algorithm.

Global optimization of generalized geometric programming

In this paper a deterministic global optimization algorithm is proposed for locatingthe global minimum of the generalized geometric programming (GGP) problem. By utilizing an exponential variable transformation and some other techniques the initial nonconvex problem (GGP) is reduced to a typical reverse convex programming (RCP). Then a linear relaxation of problem (RCP) is obtained based on the famous arithmetic-geometric mean inequality and the linear upper bound of the reverse constraints inside some hyperrectangle region. The proposed branch and bound algorithm is convergent to the global minimum through the successive refinement of the linear relaxation of the feasible region of the objective function and the solutions of a series of linear optimization problems. And finally the numerical experiment is given to illustrate the feasibility and the robust stability of the present algorithm.